Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem
Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem
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移位作用的遍历定理和 Abért-Weiss 定理的逐点版本
DOI：
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发表时间：
2018
影响因子：
1
通讯作者：
Anton Bernshteyn
中科院分区：
文献类型：
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作者：
Anton Bernshteyn
设Γ是可数无限群。遍历理论中的一个共同主题是从概率测度保持（p. m. p.）作用Γ f（X，μ）和映射f ∈ L1（X，μ），并将f的整体平均f f d μ与点态平均f <$D <$−1 ∑ δ ∈ Df（δ · x）进行比较，其中x ∈ X，D是Γ的非空有限子集.基本的希望是，当D运行在一个适当选择的无限序列上时，这些逐点平均值应该收敛到μ的全局值-几乎所有x。在本文中，我们证明了几个结果，完善了上述基本范式一致控制的平均值在特定的集合D，而不是考虑其极限为D → ∞。我们的结果包括关于Bernoulli移位作用Γ_∞（[0; 1] Γ，λ Γ）的遍历性定理，以及加强了Abért和韦斯关于移位弱包含在Γ的每个自由p. m. p.作用中的定理.特别是，我们建立了一个纯粹的Borel版本的Abért-Weiss定理的生成群的次指数增长。在我们的论证中，最近引入的可测版本的洛瓦什局部引理发挥了核心作用，这是由于当前作者以及乔卡、格拉博夫斯基、马泰、皮胡尔科和泰罗斯。
Let Γ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action Γ ↷ ( X, μ ) and a map f ∈ L 1 ( X, μ ), and to compare the global average ∫ f d μ of f to the pointwise averages ∣ D ∣ −1 ∑ δ ∈ D f ( δ · x ), where x ∈ X and D is a nonempty finite subset of Γ. The basic hope is that, when D runs over a suitably chosen infinite sequence, these pointwise averages should converge to the global value for μ -almost all x . In this paper we prove several results that refine the above basic paradigm by uniformly controlling the averages over specific sets D rather than considering their limit as ∣ D ∣ → ∞. Our results include ergodic theorems for the Bernoulli shift action Γ ↷ ([0; 1] Γ , λ Γ ) and strengthenings of the theorem of Abért and Weiss that the shift is weakly contained in every free p.m.p. action of Γ. In particular, we establish a purely Borel version of the Abért–Weiss theorem for finitely generated groups of subexponential growth. The central role in our arguments is played by the recently introduced measurable versions of the Lovász Local Lemma, due to the current author and to Csóka, Grabowski, Máthé, Pikhurko, and Tyros.
超有限性和 Borel 组合学
影响因子：
2.6
作者：
Conley, Clinton;Jackson, Steve;Marks, Andrew;Seward, Brandon;Tucker-Drob, Robin
通讯作者：
Tucker-Drob, Robin